Validated Constraint Solving—Practicalities, Pitfalls, and New Developments
Many constraint propagation techniques iterate through the constraints in a straightforward manner, but can fail because they do not take account of the coupling between the constraints.However, some methods of taking account of this coupling are local in nature, and fail if the initial search region is too large.We put into perspective newer methods, based on linear relaxations, that can often replace brute-force search with the solution of a large, sparse linear program.
Robustness has been recognized as important in geometric computations and elsewhere for at least a decade, and more and more developers are including validation in the design of their systems. We provide citations to our work and to the work of others to-date in developing validated versions of linear relaxations.
This work is in the form of a brief review and prospectus for future development. We give various simple examples to illustrate our points.
Unable to display preview. Download preview PDF.
- 1.Babichev, A. B., Kadyrova, O. B., Kashevarova, T. P., Leshchenko, A. S., and Semenov, A. L.: UniCalc, A Novel Approach to Solving Systems of Algebraic Equations, Interval Computations 2 (1993), pp. 29–47.Google Scholar
- 2.Benhamou, F.: Interval Constraints, Interval Propagation, in Floudas, C. and Pardalos, P. (eds), Encyclopedia of Optimization, Kluwer Academic Publishers, Dordrecht, 2001.Google Scholar
- 3.Berz, M.: Cosy Infinity Web Page, 2000, http://cosy.pa.msu.edu/.
- 4.Cleary, J. G.: Logical Arithmetic, Future Computing Systems 2 (2) (1987), pp. 125–149.Google Scholar
- 5.Floudas, C. A.: Deterministic Global Optimization: Theory, Algorithms and Applications, Kluwer Academic Publishers, Dordrecht, 2000.Google Scholar
- 6.Hongthong, S. and Kearfott, R. B.: Rigorous Linear Overestimators and Underestimators, preprint, 2004, http://interval.louisiana.edu/preprints/estimates_of_powers.pdf.
- 8.Kearfott, R. B.: Decomposition of Arithmetic Expressions to Improve the Behavior of Interval Iteration for Nonlinear Systems, Computing 47 (2) (1991), pp. 169–191.Google Scholar
- 9.Kearfott, R. B.: Empirical Comparisons of Linear Relaxations and Alternate Techniques in Validated Deterministic Global Optimization, preprint, 2004, http://interval.louisiana.edu/preprints/validated_global_optimization_search_comparisons.pdf.
- 10.Kearfott, R. B.: Globsol: History, Composition, and Advice on Use, in: Global Optimization and Constraint Satisfaction, Lecture Notes in Computer Science, Springer-Verlag, New York, 2003, pages 17–31.Google Scholar
- 11.Kearfott, R. B.: Interval Analysis: Interval Newton Methods, in: Encyclopedia of Optimization, volume 3, Kluwer Academic Publishers, 2001, pp. 76–78.Google Scholar
- 12.Kearfott, R. B.: Rigorous Global Search: Continuous Problems, Kluwer Academic Publishers, Dordrecht, 1996.Google Scholar
- 13.Kearfott, R. B. and Hongthong, S.: A Preprocessing Heuristic for Determining the Difficulty of and Selecting a Solution Strategy for Nonconvex Optimization, preprint, 2003, http://interval.louisiana.edu/preprints/2003_symbolic_analysis_of_GO.pdf.
- 14.Kearfott, R. B., Neher, M., Oishi, S., and Rico, F.: Libraries, Tools, and Interactive Systems for Verified Computations: Four Case Studies, in: Alt, R., Frommer, A., Kearfott, R. B., and Luther, W. (eds), Numerical Software with Result Verification, Lecture Notes in Computer Science 2991, Springer-Verlag, New York, 2004, pp. 36–63.Google Scholar
- 16.Kreinovich, V., Lakeyev, A., Rohn, J., and Kahl, P.: Computational Complexity and Feasibility of Data Processing and Interval Computations, Kluwer Academic Publishers, Dordrecht, 1998.Google Scholar
- 17.Lebbah, Y., Michel, C., Rueher, M., Daney, D., and Merlet, J.-P.: Efficient and Safe Global Constraints for Handling Numerical Constraint Systems, SIAM J. Numer. Anal., accepted for publication.Google Scholar
- 19.Neumaier, A.: Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, 1990.Google Scholar
- 20.Neumaier, A. and Shcherbina, O.: Safe Bounds in Linear and Mixed-Integer Programming, Math. Prog. 99 (2) (2004), pp. 283–296, http://www.mat.univie.ac.at/~neum/ms/mip.pdf.
- 21.Rump, S. M. et al.: INTLAB home page, 2000, http://www.ti3.tu-harburg.de/~rump/intlab/index.html.
- 22.Tawarmalani, M. and Sahinidis, N. V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, and Applications, Kluwer Academic Publishers, Dordrecht, 2002.Google Scholar
- 23.Van Hentenryck, P., Michel, L., and Deville, Y.: Numerica: A Modeling Language for Global Optimization, MIT Press, Cambridge, 1997.Google Scholar